Question

Factor the expression \(6 x^{2}+x-2\).

Short answer

(2x-1)(3x+2)

Step-by-step solution

- Identify the expression

Recognize that we need to factor the quadratic expression: 6x^{2}+x-2.

- Determine the product and sum

We need to find two numbers whose product is the product of the coefficient of the first term and the constant term (i.e., \(6 \times -2 = -12\)) and whose sum is the coefficient of the middle term (i.e., 1). These numbers are 4 and -3 since \(4 \times -3 = -12\) and \(4 + (-3) = 1\).

- Split the middle term

Rewrite the middle term (x) using the two numbers found in Step 2 to get: 6x^{2}+4x-3x-2.

- Grouping

Group the terms in pairs: (6x^{2}+4x) + (-3x-2).

- Factor by grouping

Factor out the greatest common factor from each group: 2x(3x+2) - 1(3x+2).

- Factor out the common binomial factor

Since \(3x+2\) is common in both groups, factor it out to get: (2x-1)(3x+2).

Key concepts

Factoring by Grouping

Factoring by grouping is a useful method for simplifying complex quadratic equations. To approach this, we break the middle term into two terms whose coefficients add up to the original middle term's coefficient. This makes it easier to group and factor the expression. Suppose we have the quadratic expression we need to factor: \(6x^2 + x - 2\). The process involves:

• Finding two numbers (in this case, 4 and -3) that multiply to the product of the leading coefficient and the constant term.
• Ensuring these numbers add up to the middle term's coefficient.

We then split the middle term using these numbers to create groups, which can further be factored individually. The goal is to identify and factor out the Greatest Common Factor (GCF) within each group. Through grouping, the quadratic equation becomes easier to manage and solve.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). These types of equations are essential in algebra and have various methods for solving, including:

• Factoring
• Using the quadratic formula
• Completing the square

Factoring specifically requires you to break down the quadratic expression into a product of two binomials. For the quadratic \(6x^2 + x - 2\):

• The leading coefficient (a) is 6.
• The middle coefficient (b) is 1.
• The constant term (c) is -2.

These values help determine the numbers needed for factoring by grouping. In our example, finding two numbers that both sum to 1 and multiply to -12 (the product of 6 and -2) is crucial. This makes it possible to break apart and then group the terms effectively.

Mathematical Steps

Solving a quadratic expression by factoring requires following a sequence of steps to ensure accuracy. Here's a brief outline of this process:
1. **Identify the Quadratic Expression:** Recognize what needs to be factored, as in \(6x^2 + x - 2\).
2. **Determine the Product and Sum:** Find two numbers whose product matches \(a*c\) (for \(6x^2 + x - 2\), that's 6 * -2 = -12) and whose sum matches the middle coefficient (b). For our expression, 4 and -3 fit as they multiply to -12 and add to 1.
3. **Split the Middle Term:** Rewrite the quadratic as \(6x^2 + 4x - 3x - 2\), with the middle term split into two separate terms.
4. **Grouping:** Group the terms into pairs, like \((6x^2 + 4x) + (-3x - 2)\).
5. **Factor by Grouping:** Factor out the GCF from each group; here it leads to \(2x(3x + 2) - 1(3x + 2)\).
6. **Factor Out the Common Binomial:** Extract the common binomial factor \((3x + 2)\) to get \((2x - 1)(3x + 2)\).
By systematically following these steps, we simplify quadratic expressions, making them more manageable for further algebraic manipulation.